Kovtunenko Victor A, Itou Hiromichi, Khludnev Alexander M, Rudoy Evgeny M
Institute for Mathematics and Scientific Computing, University of Graz, NAWI Graz, Heinrichstr. 36, 8010 Graz, Austria.
Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan.
Philos Trans A Math Phys Eng Sci. 2022 Nov 14;380(2236):20210364. doi: 10.1098/rsta.2021.0364. Epub 2022 Sep 26.
Mathematical methods based on the variational approach are successfully used in a broad range of applications, especially those fields that are oriented on partial differential equations. Our problem area addresses a wide class of nonlinear variational problems described by all kinds of static and evolution equations, inverse and ill-posed problems, non-smooth and non-convex optimization, and optimal control including shape and topology optimization. Within these directions, we focus but are not limited to singular and unilaterally constrained problems arising in mechanics and physics, which are governed by complex systems of generalized variational equations and inequalities. Whereas classical mathematical tools are not applicable here, we aim at a non-standard well-posedness analysis, numerical methods, asymptotic and approximation techniques including homogenization, which are successful within the primal as well as the dual variational formalism. In a broad scope, the theme issue objectives are directed toward advances that are attained in the mathematical theory of non-smooth variational problems, its physical consistency, numerical simulation and application to engineering sciences. This article is part of the theme issue 'Non-smooth variational problems and applications'.
基于变分法的数学方法在广泛的应用中得到了成功应用，特别是在那些以偏微分方程为导向的领域。我们的问题领域涉及由各种静态和演化方程、反问题和不适定问题、非光滑和非凸优化以及包括形状和拓扑优化在内的最优控制所描述的一大类非线性变分问题。在这些方向上，我们关注但不限于力学和物理学中出现的奇异和单边约束问题，这些问题由广义变分方程和不等式的复杂系统所支配。虽然经典数学工具在此不适用，但我们旨在进行非标准适定性分析、数值方法、渐近和近似技术，包括均匀化，这些方法在原始和对偶变分形式中都很成功。从广泛的范围来看，本专题的目标是朝着非光滑变分问题的数学理论、其物理一致性、数值模拟以及在工程科学中的应用所取得的进展。本文是“非光滑变分问题及应用”专题的一部分。
